Accelerating Bayesian Inference in Computationally Expensive Computer Models Using Local and Global Approximations

Computational models of complex phenomena are an important resource for scientists and engineers. However, many state-of-the-art simulations of physical systems are computationally expensive to evaluate and are black box—meaning that they can be run, but their internal workings cannot be inspected or changed. Directly applying uncertainty quantification algorithms, such as those for forward uncertainty propagation or Bayesian inference, to these types of models is often intractable because the analyses use many evaluations of the model. Fortunately, many physical systems are well behaved, in the sense that they may be efficiently approximated with a modest number of carefully chosen samples. This thesis develops global and local approximation strategies that can be applied to black-box models to reduce the cost of forward uncertainty quantification and Bayesian inference. First, we develop an efficient strategy for constructing global approximations using an orthonormal polynomial basis. We rigorously construct a Smolyak pseudospectral algorithm, which uses sparse sample sets to efficiently extract information from loosely coupled functions. We provide a theoretical discussion of the behavior and accuracy of this algorithm, concluding that it has favorable convergence characteristics. We make this strategy efficient in practice by introducing a greedy heuristic that adaptively identifies and explores the important input dimensions, or combinations thereof. When the approximation is used within Bayesian inference, however, it is difficult to translate the theoretical behavior of the global approximations into practical controls on the error induced in the resulting posterior distribution. Thus, the second part of this thesis introduces a new framework for accelerating MCMC algorithms by constructing local surrogates of the computational model within the Metropolis-Hastings kernel, borrowing ideas from deterministic approximation theory, optimization, and experimental design. Exploiting useful convergence